Logarithmic scale



C. L. RAGOTET AL LOGARITHMIC SCALE FiledJuly 9. 1923 Dem-14 1926.

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GF NEW YORK, N. Y.

LO-ARITHMIC SCALE.

Application iilcd July S The ordinarv live )lace losarithmic tables 1 Cor ei'nteen pages Kcuve the l illglnn to live plac .i tor each tourplace nber from 1,000 to 9,909. To lind the oa 'ithm tor any live placenumber, it necessary to interpolate, that is, to add to the logarithm ofthe four place number correspondingij to the lirst tour di its ot thelive place number, the proper proportionate part ot the dinerencebetween it and the logarithm of the next successive tour place number.This includes subtraction to obtain the difference, multiplication toobtain the amount to be adrled, and addition.

ln usino; such a table there are many sloiuces oi' error. lheie iii-aybe a typogrraphi error in the prin-tim;y oit the table, and i ht beimpossible to detec this from an inspection of the table. lai-lv theerror be in the last ci; i

The user may tail t Yhe roiv of figures across the eager. Aicularl Pdi/Jitoii the number y iii the fourth lim is required be a El or l Ythinprint '3 r nay tuus talfe the l. above or belenT the one van lle maymisread the logarithm, ii ce reaciim` a 3 tor an S, pri'icular rint orthe light be poor.

most prolific sources of error i A on. To lind the anti-loggan rithm,that is. the number cor= sponding to a given logarithm, is diliicult,the logarithm one is looking` for, even it oi' but four figures isgenerally not in the table, and one must do a more difficult form olinterpolation to find the anti-logarithm.

To avoid the necessity for such interpolation one may use tables givingthe numbers to tive places, but such tables cover ten times as manypages on the tables giving; the numbers to four places, and aretherefore very much more costly and more bulky to handle. Furthermore,they require a longer time in turning` the pages to locate the numberdesired.

One of the main objects of our invention is to provide, in compact andclearly readable lform, means for the direct reaoinfg` ol? numbers tolive places Without the need for interpolation, and in which sources oterror in printing will be readily apparent to the most casual glance,and error in locatingl or readingl is reduced to the lowest possibleminimum.

In carrying out our invention there is 1923. Serial No. 650,223.

provided a scale graduated to represent all ot the numbers from 10.000to 100,000, that 1s, 'for all of the numbers oi'tive places, andjuxtaposed to this scale is a second one subdivided into spacesrepresenting` the entire range from the logarithm ot 10,000 to thelogarithm of 100,000, constituting a complete logarithmic cycle. Thenumbers are therefore indirect sequence in both scales, but the spacingis sch that the logarithm of any number in one scale is directly opposite to said number and on the other scale. Logarithms oit all numbersof tive places may be directly read Without interpolation, and as everylogarithm is present up to five. places, it is just as easT7 to find thenumber Corresp(aiding,l to any logarithm as it is to find the logarithmcorresponding .to any number. is all of the figures in the numbei` scae, as well in the logarithmic scale, runl indirect sequence Withoutomissions, the finding or" logarithme or numbers is very easy, and thereis little or no danger of mistake. Y

@ur improved scale might be in the forni ot a single strip or tape whichmight be mounted on a pair of parallel spools and 'wound from one toanother to brine; any partinto view, or it might be printed in helical'form on the surface ot' a drum. Preferably itis subdivided intosections arranged parallel and in succession on suc-- cessive sheets,'with several sections per sheet.

ills the dille-rence between logarithme for successive numbers decreasesas the numbers increase, a further object of the invention is to so formthe scale that the numbers and logarithms may be read With equalfacility at all parts oit the scale. To accomplish this object the sctions at different points along the scale are made With the numericalgraduations at different dis'ances apart. F or instance, the rst sectionof the scale may have only thirty numbers, Whereas the last section ofthe scale may have one hundred and twenty-live numbers, although thesections are of equal length. This iirst section Will have one hundredand thirty logae 1itlnnic graduations, the logarithm of 10030 is 00180.(disregarding the decimal points). The last section will have only aboutfifty-four subdivisions in the logarithmic scale, as the logarithm of99,875 (100,000 minus is 09,94, (100,000 minus 54). Intermediatesections ot the same length may have Jiorty, titty, sixty, eighty,

or one hundred numb if, as the diiierence between the logarithme ofsuccessive numbers decreases. The selection oi the proper spacing ofnumber per secton s such that at no point ni the scale so close togetheras to pie irom being easily read.

is another important 'eature oi our invention, all ot Jdie scalesections are arranged at right angles to arpair oi columns, onecontaining two leadil g digits o'i the numbers, and the other containingthe two leading digits oi the logarithme. Yl`he successive numbersconstituting these leading digits are spaced along the column vso tocome opposite to each thousand subdif'isions on the scale sections. lnother w rds, l as the two leading digits of 10,000, would appearopposite the beginning the tiret scal section, and would` not berepeated except possibly at tl e i koi' a new page. ll the ytwo leading's ot 11,090 would. appear opposite the aeginning ot the scale sectionwhich has the thousandth subdivision from the beginning. OO would occurin the column for logarithme and opposite to the beginning or the .firstsection, and would not be repeated except possibly at the top oi n iage,while 0i would appear in the logariti the beginning olf the sectionhavii. thousandth subdivision in sae scale. rlhe thA l and t'ourthdigits oi l. the numbers i logarithme printed 'succession along thesuccessive scale sections land at the proper spacing, while the Frithdigitoi: the number and ol thelogarithni is not printed but is indicatedby the ten subdivisions along the'scale sections between successivepairs ot digits on the latter.

To find any number it is merely necessary to go down the number columnto reach the first two digits, then across the successive scale sectionsto reach the nent two digits ot the number, and then count subdivisionsalong the scale equal to the last digit ot the number. rlhis point iscarefully noted. rlhe logarithm corresponding to the number may be readdirectly. lts first two digits will be in the logarithm column beforethe particular scale section containing the located point, its next twodigits will be on the scale section on the logarithm side in advance otthe point located, and its filth digit will be the number or" thelogarithm scale divisions between the last mentioned pair ot digits andthe graduation nearest to the located point lt will be obvious thatJfinding anti-logarithms involves exactly the same process and isexactly as easy, and neither process involves any interpolation. fis thefirst two digits of the number, that is, 0 to 99, .are printed insuccession down the number column, and the lirst two digits ot thelogarithm, that is OO to 99 are printed in succession down the logarithmcolumn, and as the numbers OO to 99 occur in succession along the scalesections between each two consecuti've numbers in the nui'nber columnand also between each two consecutive numbers in the logarithm column,it Yill be apparent that prooi reading to detect any typographical errorin the printing is very simple. ldjacent numbers must be consecutiif'enumbers in both columns along the scale, and ror is very easilydetected. rihis also equally tacilitates the reading ot' numbers, theuser is not liable to mista-he one number lfor another. For instance, ita 9? be poorly printed so to look like a Si, he will readily notice thatit should be a 9?, as it occurs between 96 and 98. 'lhis is not the casewherV there is a big jump in numbers, as in the logaritl A'ns printed inthe ordinary table.

in the accompanying drawing we have illnstrated certain et the scalesections which. may be mployed in making up a completo scale. ln thesedrawings:

l represents the lir t section ot the scare.

is a grouo et intermediate sections, and

Fig. 3 is the last section.

lt will be noted t` at each section includes a line ifi o' convenientlength to be printed onan oi linary page, and this line intersects twovertical columns and C which are headed Nos for nun bers and"liogs. torlogarithins. fit the upper side ot each lino fi are a series oi")divisions uniformly spacedvfor each section but the spacing need not bes e tor dii'lerent sections. Each tenth subdivision is numbered, andeach pair along these subdivisions enc c 5 may be undii" to nilitatcipid ocation et any particular number. ielow each line n, are a serresot subdivisions non-unijformly spaced, the spacing being according tothe logarithmic progression, and completing one logarithmic cycle. FromFig. l it will be seen that there are one hundred and thirtysubdivisions on the logarithmic )art ol" scale for thirty subdivisionsin the nuniber part o'i the scale, whereas in 'l' i 3 will be seen thatthere are titty-tou subdivisions opt the logarithmic scale 'lor onehundred and twenty-live subdivisions ot' the number sc in neithersection is there more than one hundred and thirty subdivisions in eitherscale. li course it the scale be printed on larger page there may be asmaller number ot sections, and each section may have larger number otsubdivisions tor both scales.

- lt will be apparent Vthat the change b,.- tween successive thousandthswill lrequently occcur in the logarithmic scale intermediate ot the endsof a section, and any suitable means may be employed i'or indicatingthis and lll() lill .fact and instructing the user to take the nextlower number in the logarithmic column rather than the preceding numberin said column, to obtain the lirst two digits et the logarithm. Thesame applies to the number scale, although the scale sections may bcmade ot such length that this change does not occur so often. As aconvenient manner of indicating this change we employ a second line Dparallel to the line A and between it and the numerals of the scale, andextending from the 00 to the end of the scale section, wherever such 00occurs intermediate of the ends of either the number scale or thelogaritlnnic scale.

The following are a few examples ol" the use of our improved device.

lf the logarithm of 10,024 is desired, the user locates the first twonumerals, namely 10 in the number column, then goes across the scale tothe numerals 02 in the iirst scale section, which are the third andfourth digits of the number, yand then counts to the fourth subdivisionon the scale aiter 02. This located point is marked ai in Fig. 1 of thedrawing. The logarithm ot this number will be 00104. Ol? the logarithm,the lirst two digits are in the Column C, the next two are the numeralbelow the line A and in advance or the located point ai and the lastdigit is the fourth subdivision beyond the numeral 10.

The linding of a number corresponding to any logarithm is equallysimple. For instance, the number corresponding to the logarithm 4:3025is found by locating the point which is marked y in Fig. 2. Because the00 occurs in the logarithm scale in the same line with 025, the numeral:"'or the lirst two digits is opposite the line below. The numbercorresponding to this logarithm will be 20931.

The scale is veryinuch more accurate than the ordinary live placelogarithmic tables. f

ln such tables the last digit is an approximation. For instance, thelogarithm for 9990 is given in the table as 99957. From the scale (F ig.3) it will be noted that the logarithm of 99901 is 99957 and thelogarithm ot 99899 is 99956. The scale therefore shows that thelogarithm ot 9990 is actually 999565. In F ig. 3 the numerals 99 are notshown in either the column B or the column C as this ligure shows onlythe very last line of the scale, and the numerals 99 in the 'two columnswould be farther up on the page bearing this last line.

The specilic construction disclosed in the accompanying drawingsembodies Briggs logarithms of the natural numbers on the base 10. Itwill of course be evident that the invention is also ap olicable tologarithms with other bases than 10, the logarithms of the trigonometricfunctions such as log sin, log cos, log tan, log cot, etc. lt may' alsobe used ior giving the natural functions oi the angles, such as sines,cosines, cotaugents, secants and cosecants. lt may also be used for thedirect reading of values, particularly where the progression of valuesis other than a uniform one, as ior instance in determining thecircumference, area, spherical volume, capacity in gallons, liters, orthe like, ytor each successive change in diameter or other function.

it will be readily seen that our invention makes possible the embodimentot otherwise elaborate, voluminous and c nnbersome tables into a smallcompass of convenient and compact iorm which is at the same timereadable from either scale to the other with reliability and accuracy.

. ln the drawings, the scales given are for numerals of tive places andthe logarithme may be directly read to live places, and in some caseclosely approximated to the sixth place. lt will ot course be evidentthat the same character ot' table may be used for siX place tables inwhich case the lirst three numerals would appear in the number columnand in the logarithm column, instead of only the iirst two, shown inthe` drawing.

ln all ot the nples hereinbefore given, and in the description thequestion ot decimal point has not been taken into consideration, as thesaine rules for placing the decimal and for determining thecharacteristic before the decimal point which apply in the use ellogarithmic tables will apply in the use oi? our scale.

Having thus described our invention, what we claim as new and desire tosecure by Letters Patent is:

1. A Chart comprising a group of horizontally disposed parallel linesarranged adjoining to a pair of `vertically disposed marginal columns atthe left hand margin, said lines having a continuity of graduationsbeginning at the margin ot' the lirst line and ending at the end of thelast line, the graduations being arranged in two series and extending inopposed relation to each other along` the said lines, one of said seriesot graduations being spaced in aritlunetical progression and bearingsuitable numerical indicia readable partly in one of said marginalcolumns and partly along' said graduations, the other orP said series ofgraduations being spaced in logarithmic progression and having similarindicia readable partly in the other of said marginal columns and partlyalong the second said graduations, and indicia disposed at the top ofthe said marginal columns to denominate the values of the indicia.thereunder.

2. A chart comprising a group of parallel lines arranged adjoining to apair ot' marginal columns, said lines having a continuity tangeuts,

or graduations beginning at the margin ot the iirst line and ending atthe end of the last line, the graduations being arranged in two seriesand extending in opposed relation to each other along the said lines,rone oit said series of gradnationsV being spaced in one order ofprogression, and bearing suitable numerical indicia readable partly inone of said marginal columns and partly along said graduations, theother ot said series or graduations `being spaced in another order otprogression and having siinilar indicia readable partly in the other otsaid marginal columns and partly along the second said graduations, andindicia dis posed at the top of the said marginal eoluinns to denominatethe values oit the indicia thereunder.

3. A chart comprising a continuity ot two adjoining scales plotted withgradnations on opposite sides of a horizontal line., each being readablein terms ot the other, said line being sectionizedY in convenientlengths, said sections being groi-iped in consecutive order along amargin having two columns intersecting said line, the graduations oitone scale being plotted in equal increments, an d the graduations of theother scale being plotted in varying increments, indicia consisting ofdigits, indicating numerical values or quantities and snbdiw'isionsVthereof, saine being readable partly in one ot said inarginal columnsand partly along the equally spaced graduations, similar digitsindicating other values, being readablepartly in the other of saidmarginal columns and partly along the graduation's ot varyingincrements, and indicia disposed at the top or" said marginal columns todenominate the digits thereunder.

et. A chart comprising a scale, in which two adjacent series otgraduations bearing a iixed relation to each other are sectionized inconvenient lengths arranged to begin at common margin, separatenumerical indicia indicating the values of each or" said series oladuations, two columns in said 11a 5 part ot the digits oil'. each o lnidicia, the remainder ol each of said numeri-cai indicia being readablealong each of the said series ot graduaions, the graduations ot' one otsaid series being spaced to represent numerical values or quantities,the graduations of theV other oit said series being spaced to representthe logaritliins corresponding to the numerical values or quantitiesrepresented nations of ythe other series, the values indicated in eachseries being readable in terms ot 'the other. 1

i chart having a pair of vertical colninns acent to thev lett handmargin, and a group ot horizontal lines extending to the rom saidcolumns, said lines having innity of graduations beginning at left handend oi the irst line, and endat the right hand end of the last line,Jdnati ns beingarranged in two series pon opposite sides oli each line,one of said i' dnations being` spaced in aritl inetical piogiession andbearing numerical the first two digits oiu which are sed in one of saidmarginal columns d the third and fourth digits of which are "sa edpoints along said lines, the other ot a fies of graduations being spacedin 'thniic progression and having siinila' ia, the iii-st tivo digits ofwhich are in the other or" said marginal columns and the second andthird digits being at spaced points along the second ot saidgradnations. Signed at New York, in the county of lew York and State or"ll ew York, this 2nd day or duly, 1923.

CHARLES L. RAGGT. ADREN LACROl/.

by trie grad-`

